From a Floyd–Auslander minimal system to an odd triangular map

Abstract

In 1989 A.N. Sharkovsky asked the question which of the properties characterizing continuous maps of the interval with zero topological entropy remain equivalent for triangular maps of the square. The problem is difficult and only partial results are known. However, in the case of triangular maps with nondecreasing fibres there are only few gaps in a classification (given by Z. Kočan) of a set of 24 of these conditions. In the present paper we remove these gaps by giving an example of a triangular map in the square with the following properties: (1) all fibre maps are nondecreasing, (2) all recurrent points of the map are uniformly recurrent, and (3) the restriction of the map to the set of recurrent points has an uncountable scrambled set (and so is Li–Yorke chaotic). The example is obtained by taking an appropriate Floyd–Auslander minimal system and then taking its appropriate continuous extension to a triangular map of the square.